Understanding Vector Spaces, Linear Algebra, Eigenvalues, and Quadratic Forms

Posted on Mar 11, 2025 in Mathematics

A.1.7. Vector Subspaces

A subset H of a vector space V is a vector subspace if and only if:

  • The zero vector 0 is in H.
  • For any vectors v1 and v2 in H, their sum v1 + v2 is also in H.
  • For any vector v1 in H and any scalar k, the scalar multiple k*v1 is also in H.

Examples of non-vector subspaces:

  • Sets defined by polynomial equations.
  • Vectors u = (x, y) where xy + x = 0.
  • Logarithmic functions.

Examples of vector subspaces:

  • Sets defined by linear equations like Ax + By + Cz = 0.
  • Sets defined by linear equations like C…X + …Y = 0.

A.4.2. Kernel and Image of a Linear Transformation

Let f be a linear transformation from E to L.

Kernel

The kernel of f, denoted as ker(f), is the set of all vectors in E whose image under f is the zero vector in L. In other words, ker(f) = {x ∈ E | f(x) = 0}. The kernel of f is a vector subspace of E.

Image

The image of f, denoted as im(f), is the set of all vectors in L that are the image of some vector in E. In other words, im(f) = {y ∈ L | ∃x ∈ E, f(x) = y}. The image of f is a vector subspace of L. The dimension of the image coincides with the rank of the matrix associated with f: dim(im(f)) = rank(M(f)).

Important Properties

  • f is injective if and only if ker(f) = {0}.
  • f is surjective if and only if rank(M(f)) = dim(L).
  • dim(ker(f)) + dim(im(f)) = dim(E).
  • If E = L and f is injective, then f is bijective.
  • If E = L and f is surjective, then f is bijective.

A.5.1. Eigenvalues and Eigenvectors

A scalar λ (lambda) is an eigenvalue of a matrix A if there exists a non-zero vector x such that:

Ax = λx

If λ is an eigenvalue of A, then the non-zero vectors x that satisfy the above equation are called eigenvectors of A associated with λ.

Each eigenvector is associated with a single eigenvalue. However, each eigenvalue can be associated with one or more eigenvectors.

A.6.1. Quadratic Forms

A real quadratic form Q on the vector space Rn is a mapping:

Q: Rn → R

such that Q(x) is a real number and satisfies the property Q(λx) = λ2Q(x) for any x in Rn and any scalar λ.

Also, Q(0) = 0.

Classification of Quadratic Forms

  • Positive Definite (PD): If Q(x) > 0 for all non-zero x.
  • Positive Semidefinite (PSD): If Q(x) ≥ 0 for all x.
  • Negative Definite (ND): If Q(x) < 0 for all non-zero x.
  • Negative Semidefinite (NSD): If Q(x) ≤ 0 for all x.
  • Indefinite: If Q(x) takes both positive and negative values.

C.1.1. Contour Lines

A contour line consists of all points where a function has the same value. Given a function f(x, y), set it equal to a constant value, and express y as a function of x. The resulting curve represents the contour line.

C.1.2. Continuity

For f(x):

f(x) is continuous at a point if the left-hand limit and the right-hand limit at that point coincide with the function’s value at that point.

For f(x, y):

f(x, y) is continuous at the point (x0, y0) if the limit of f(x, y) as (x, y) approaches (x0, y0) coincides with the function’s value at (x0, y0).

Properties of Continuity:

  • All elementary functions (polynomials, sine, cosine, exponential, logarithm, etc.) are continuous throughout their domains.
  • Elementary functions combined through addition, subtraction, multiplication, or division result in a continuous function (where defined).